Are you concerned about OTB chess or playing online ? In online chess ( turn based ) you are allowed to use books and data bases so you dont need to memorize opening moves at all.
How many opening moves can/should you remember?

I remember hundreds of opening moves. The hard part is getting them in the right order with the right system.
Reb is right, in online/correspondence chess use of opening manuals and databases can extend the "opening" phase/line into 20+ moves in the most heavily analyzed lines. OTB people may know their main line 10-15 moves deep but if there is an early divergence get lost. That is why "trick", offbeat lines are so popular at the lower levels.
At the higher levels players know more lines, better and professionals tend to be well booked. When they get hit by an odd line, especially by novices, they rely on principles, experience and superior ability to offset lack of knowledge on a specific opening.

i think about 30 openings would be enough, although even players up to 1800 don't fully know the ruy lopez
Berlin Defence
Bird's Defence
Breyer Variation
Centre Attack
Chigorin Variation
Classical/Cordel Defence
Closed Defence
Exchange Variation
Marshall Counter-Gambit
Marshall Counter-Gambit, Avoiding the
Morphy Defence
Open Defence
Schliemann/Jaenisch Gambit
Smyslov/Zaitsev Variation
Steinitz Defence
Worrall Attack
for an example there

In OTB chess its important to know your openings well, to save time on the clock if for no other reason. The faster the time control the more important it is imo. If you know the openings you play most often 8 to 12 moves deep it should be good enough. Beyond that and you are really getting into middlegame territory more often than not. Remember the only goal in the opening is to reach a playable middlegame. Reshevsky was one of the worlds strongest players and he was notorious for not knowing the openings very well ( for his level ! ) and yet he usually got through them ok BUT he very often was in time trouble because he was having to find at the board theoretical moves that his opponents knew and were playing much faster.

i don't know, learning openings on your own is good in my oppinion, because it trains you to see stuff you might not see if you've had the best lines spoon fed to you. also, it keeps you from being too taken aback when someone plays an opening that isn't common, because you don't know that it isn't common!
The main openings I play I probably know 10-15 moves out. Unless it is an extremely forcing opening where a minor transposition loses knowing the general principles is more than enough--especially when you are confronted with a side line that is new to you.
In correspondence play I played a King's Indian where we followed book for 24 or so moves--not that unusual.

Ofcourse, you should understand the opening moves you are making and not just memorize moves that you dont understand. Chess is a game of understanding, not memory. - Korchnoi

I'd say between a few hundreds and a few thousands for competitive amateur OTB players.
IMs and GMs probably know much more, and super-GMs, I just can't figure it.
btw, I'm counting "positions" here, like in a tree repertoire book, ie :
1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Bxc6 counts as 4 moves for white
1.e4 e5 2.Nf3 Nc6 3.Bb5 d6 4.d4 exd4 5.Qxd4 counts as an extra 2 moves for white
and so on...

To know your openings is an important part of the game. An year ago, I nearly didnt know an opening. Now I know and more important I understand my openings, now I play on a higher level. I play on this higher level because I have better playable middlegames. It's much easier to play in an equal or better position than fighting against a better position of your opponent.
But dont learn them, understand them! And you dont have to know 15-20 moves, normally 8-12 moves are more than enough. But this depend on your level and your opponents level.

It's good to look at different openings and to use the ones that your most comfortable with, after that learning as much as you can about the openings you favor should only help your game even if it's one.

Thanks for the link kingforce (#4). Studying it methodically will take a long time, but I have to start somewhere.
My online chess rating is 1805, and I only know one opening.
Your rating corresponds to 1450-1500 USCF, it's class C, not that much. If you want to go higher, at least to class "B" ( i.e. ~2000 here) you must know at least 8-10 openings. There are a lot of games that are lost in 10-15 moves just because of not knowing the opening.
I personally know around 17-20 openings somewhat well, at least 3 moves in. I only use around 4, but still...good to know what's out there, even if you don't use it.

Thanks To HUNgarian for starting the topic & others for the valuable discussion so far!
In total,how many openings are there?

Thanks To HUNgarian for starting the topic & others for the valuable discussion so far!
In total,how many openings are there?
It depends how you define an opening : 1.e4 e5 2.Nf3 Nc6 3.Bb5 is the Ruy Lopez, now, if Black answers 3...a6, it's called Morphy defence, and if white replies 4.Bxc6, it's called Exchange variation, and it goes on and on...
As you can see, major openings are subdivided into variations and sub-variations, and so on and so forth...
But if we're talking about major openings, I'd say roughly 50.
I find it amazing when some of the instructional articles casually mention that for instance move 14. b5 is more popular nowadays.
So, I am wondering how many different openings and up to how many moves should a 1600 and a 2000 rated player memorize.
And also how many opening moves do players actually know?